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JOHN CASTI has been widely praised for his two previous books, Searching for Certainty and Paradigms Lost. Justly so, for he has a knack of explaining difficult concepts with unusual and entertaining ease. A mathematician at the Santa Fe Institute, Casti has a clarity of thought and vision that is reflected in his writing. Complexification is in the same genre as his other two books. 鈥淸Both] natural and human affairs are just plain complex,鈥� he writes.

For reasons having to do with the Western scientific tradition 鈥� that the best and truest theories and natural laws are the simple ones 鈥� and, probably, the human brain鈥檚 yearning for simplicity, our intuitions about the world are turning out to be wrong the closer we examine it. Casti portrays this paradox with wit and incisive examples. He says he is urging 鈥渁 call to arms in this battle with the complex鈥�. Those who would like to be spectators of the coming skirmishes should read this book.

It is not for the faint-hearted, however, because Casti does not shrink from some mathematics that many will find challenging. And in parts it reads like a text for scientists, not a book for the general public. 鈥淐onsider a bundle of light rays emanating from a point source,鈥� a section on physics and metaphysics begins. He asks the reader to consider many such situations throughout the book, and the effort is well worth it. In any case, Casti leavens the text with many anecdotes, some from his own experience. Through his book we learn that the world is a surprising place, and begin to see ways of understanding it.

There are five main sections, each of which opens with a statement of our intuition, which, it transpires, is wrong. The first section, for example, is called 鈥淭he catastrophic鈥�, and the intuition is that 鈥渟mall, gradual changes in causes give rise to small, gradual changes in effects鈥�. Not so, we learn. From mass extinctions, to the collapse of the stock market, to the rise and fall of civilisations, events are seen to follow anything but a simple cause-and-effect relationship.

In the section headed 鈥淭he chaotic鈥�, we see that the intuition that 鈥渄eterministic rules of behaviour give rise to completely predictable events鈥� is violated everywhere, from weather systems to ecosystems. 鈥淲atching nature red in tooth and claw is certainly one way to see counterintuitive, chaotic behaviour rear its ugly head,鈥� writes Casti. Such behaviour is to be found in 鈥渆very nook and cranny of day-to-day life鈥�, so that 鈥渢he universe is incapable of predicting the future behaviour of even a small part of itself鈥�.

The third section is 鈥淭he lawless鈥�. The intuition here is that 鈥淎ll real-world truths are the logical outcome of following a set of rules鈥�. Surprise 鈥� some things, perhaps including human creative thought, are beyond simple computation. Similarly, in the fourth section, 鈥淭he irreducible鈥�, we are told that, contrary to common intuition, some complicated systems cannot be understood by breaking them down into simpler parts. In this section, the reader will be surprised to encounter, side-by-side, discussions of ecosystems, A Midsummer Night鈥檚 Dream, and the geometry of chess. That鈥檚 Casti for you. He sees the connections and, in the end, so does the reader.

The last section, 鈥淭he emergent鈥�, is the core of the science of surprise, as Casti calls it. Here the intuition is: 鈥淪urprising behaviour results only from complicated, hard-to-understand interactions among a system鈥檚 component parts.鈥� Once considered somewhat mystical and beyond the realms of real science, emergence is now seen everywhere in both the physical and biological worlds. The classic example here is John Horton Conway鈥檚 cellular automata, which, from simple rules, produce complex and unpredictable patterns. It is a metaphor for the natural world in all of its guises, and the lesson we learn, says Casti, is that we should 鈥渆xpect the unexpected鈥�.

Casti ends Complexification with a discussion of what we mean by the complex and complexity. He also discusses how our usually anecdotal experience of such phenomena might be transformed into a real science. Complexity, he argues, is a subjective phenomenon: 鈥淲hat鈥檚 complex depends on how you look.鈥� Although it is true that no system operates in isolation, which affects levels of complexity that may be relevant, some systems are, if such a phrase may be allowed, irreducibly complex. Casti鈥檚 claim that complexity is subjective is, therefore, beside the point, although he is right to point to the confusion that arises with the use of the same word 鈥渃omplex鈥� in anecdotal and formalistic contexts.

This is not the first book on complexity, nor will it be the last. Much of the background to the topics Casti pulls together is to be found in an extensive and extremely useful 鈥淒ig deeper鈥� section. This is effectively a bibliography for those who wish to come to grips with what is surely more than just the passing fantasy of a few creative people鈥檚 overcharged minds.